Galois Representations and Modular Forms

伽罗瓦表示和模形式

• 批准号: 1252158
• 负责人: Richard Taylor
• 金额: $ 64.11万
• 依托单位: Institute For Advanced Study
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1252158&HistoricalAwards=false
• 关键词: Galois; Representations; Modular; Forms; Galois; Representations; Modular; Forms

A big theme in number theory in the last 50 years has been the relationship between automorphic forms, Galois representations and objects from algebraic geometry. There is an extensive web of extraordinary conjectures (for instance the Artin conjecture, the Shimura-Taniyama conjecture, Langlands' conjectures,Serre's conjecture and the Fontaine-Mazur conjecture) linking these three seemingly very different subjects (which relate to analysis, algebra and geometry respectively). Progress on these conjectures is currently very exciting. Under previous NSF grants the PI, with various collaborators, completed the proof of the Shimura-Taniyama conjecture; proved the local Langlands conjecture for GL(n) over a p-adic field; proved the Sato-Tate conjecture for elliptic curves over totally real fields; proved the first general automorphy lifting theorems and potential automorphy theorems for Galois representations of arbitrary dimension; and proved that the L-function of any polarized, regular, irreducible motive over a CM field has meromorphic continuation to the whole complex plane and satisfies the expected functional equation. The PI proposes to continue to improve the currently available automorphy lifting and potential automorphy theorems; to relate the cohomology of Rapoport-Zink spaces to the local Langlands conjecture for groups other than GL(n); with Kevin Buzzard and Joe Rabinoff to prove the Artin conjecture for odd degree two representations of the Galois group of a totally real field in which 5 splits completely; and to think about more speculative problems relating Galois representations and automorphic forms, for instance how to understand the case of very degenerate Hodge-Tate numbers/infinitesimal character. In addition the PI will continue his work with post-docs and, particularly, with graduate students.This circle of ideas is the one that led to Andrew Wiles' celebrated proof of Fermat's last theorem after over 300 years. They fall into the general area of arithmetic geometry - a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.

在过去50年中，数字理论的一个重大主题是自动形式，galois表示和代数几何形状的对象之间的关系。有一个广泛的猜想（例如，Artin猜想，Shimura-Taniyama猜想，Langlands的猜想，Serre的猜想和Fontaine-Mazur猜想），将这三个主题（分别与分析，代数和几何形式相关）。这些猜想的进展目前非常令人兴奋。 在先前的NSF下，PI与各种合作者一起完成了Shimura-Taniyama猜想的证明。在P-Adic领域证明了GL（N）的当地Langlands猜想；证明了对完全真实领域的椭圆曲线的sato-tate猜想；事实证明，对于任意维度的GALOIS表示，将第一个通用自动形成定理和潜在的自动形理定理；并证明了CM场上任何极化，规则，不可还原动机的L功能都可以延续到整个复杂平面并满足预期的功能方程。 PI建议继续改善当前可用的汽车提升和潜在的汽车定理；为了将Rapoport-Zink空间的共同体与GL（n）以外的其他群体相关联的兰兰兹猜想；凭借凯文·巴扎德（Kevin Buzzard）和乔·拉比诺夫（Joe Rabinoff），证明了对奇数程度的Artin猜想，这是一个完全真实领域的Galois组的两个代表，其中5个完全分裂。并考虑有关GALOIS表示和自动形式的更投机性问题，例如如何理解非常退化的Hodge-Tate数字/无限特征的情况。此外，PI将继续与博士后，尤其是与研究生一起工作。这一想法是导致安德鲁·威尔斯（Andrew Wiles）在300多年后获得Fermat的最后定理证明的想法。它们属于算术几何形状的一般领域 - 一个融合了数学最古老的领域的主题：数字理论和几何形状。事实证明，这种组合非常富有成果。其许多后果包括新的错误纠正代码。 此类代码对于现代计算机（硬盘）和紧凑磁盘都是必不可少的。